Short answer: no
Long answer #1: probably not
PCIe slot in A1222 is x16 mechanically, but only x4 lane electrically. And you must divide it to two ( or more ) parts.
As bifurcation of PCIe port not supported now in firmware, you need PCIe switch and not PCIe splitter. I.e. PCIe card with onboard switch and PCIe connectors.
Switch has to be capable for switching 4 lanes, ideally PCIe x4 to 2x PCIe x4 or PCIe x4 to 2x PCIe x2.
I am not sure, if such cards exists.
Alternativelly there are many switches PCIe x1 to 4x PCIe x1, but with these speed to graphics card will be very limited. Not recommended.
And of course, Catweasel is PCI, not PCIe. It means you need additional PCIe-PCI bridge. And even if it will works ( probably not, switch+bridge, is too complex for firmware), you ended with real monster

and not fit to Mini-ITX case.
Long answer #2: maybe in future
CPU P1022 has upto three internal PCIe roots. So it can be configured to support this combinations:
i) PCIe x4 .. this is A1222 firmware setting
ii) PCIe x2 x2
iii) PCIe x2 x1 x1
If sometimes in future U-Boot will be upgraded and will support bifurcation, you can connect simple PCIe splitter and use two or three PCIe cards.
And PCIe-PCI bridge applies here too.